Instituto de Matemática Pura e Aplicada

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1 Instituto de Matemática Pura e Aplicada Doctoral Thesis MINIMAL AND CONSTANT MEAN CURVATURE SURFACES IN HOMOGENEOUS 3-MANIFOLDS Ana Maria Menezes de Jesus Rio de Janeiro 2013

2 Instituto de Matemática Pura e Aplicada Ana Maria Menezes de Jesus MINIMAL AND CONSTANT MEAN CURVATURE SURFACES IN HOMOGENEOUS 3-MANIFOLDS Thesis presented to the Post-graduate Program in Mathematics at Instituto de Matemática Pura e Aplicada as partial fulfillment of the requirements for the degree of Doctor in Philosophy in Mathematics. Advisor: Harold Rosenberg Rio de Janeiro August 2nd, 2013

3 Aos meus pais, Antônio e Vivalda.

4 Agradecimentos A Deus, por me permitir alcançar esse objetivo, por colocar no meu caminho pessoas muito especiais, incentivadoras desse meu sonho. Ao meu orientador, Professor Harold Rosenberg, exemplo de pessoa dedicada à matemática, que ama o que faz, e que transborda esse seu entusiasmo para seus alunos. Muito obrigada pelas conversas sempre animadoras, pela paciência em explicar cada detalhe, pelo incentivo de todos os dias: Coragem!. Me sinto honrada em ter sido sua aluna. Aos Professores Manfredo do Carmo, José Espinar, Frank Pacard, Walcy Santos e Detang Zhou, por participarem da banca de defesa de tese. Obrigada pelas valiosas sugestões, e pelo interesse em meu trabalho. Um agradecimento especial ao Professor Manfredo do Carmo, por todas as conversas incentivadoras, por me apoiar em todas as fases do doutorado. Nunca esquecerei do dia anterior ao meu exame de qualificação quando ao me ver no corredor percebeu de imediato o meu nível de estresse e me convidou para ir a sua sala, para uma conversa muito agradável e tranquilizadora. Muito obrigada, Professor Manfredo! O senhor é um exemplo a ser seguido. Ao meu querido Professor Elon Lages Lima. Assim como muitos outros estudantes brasileiros, aprendi a apreciar a matemática através dos seus livros. Obrigada pelas conversas incentivadoras desde os meus primeiros dias como aluna no curso de verão no IMPA. Obrigada pelo sorriso de Vá em frente! quando me encontrava nos corredores, sempre preocupado em saber como eu estava. Muito obrigada por me considerar sua conterrânea, a alogoana aqui sente-se honrada. Aos meus pais, Antônio e Vivalda, meus maiores incentivadores. Nunca esquecerei de suas palavras que ouvia quando ainda criança: É através dos estudos, minha filha, que se vence na vida!. Obrigada por abdicarem de iv

5 tantas coisas, por trabalharem dobrado a fim de proporcionar a mim e a minhas irmãs condições para estudar. Obrigada por sonharem junto comigo. Hoje podemos dizer: Conseguimos! Às minhas irmãs, Vilma, Aline e Beatriz. Meus amores, minha fonte de alegria. Às minhas avós, Nininha e Zefinha. Aos meus tios e primos. Obrigada pelo apoio de sempre. Aos meus avôs, José e Jõao Francisco. Tenho certeza que estão lá de cima vibrando com a conquista dessa neta que sente muito a falta deles. A todos do grupo de Geometria Deferencial do IMPA. Aos meus queridos amigos Alessandro Gaio, Almir Santos, Ana Clara Noronha, Bruno Gois, Elaís Malheiro, Ivaldo Nunes, Jyrko Correa, Lucas Ambrozio, Lucas Backes, Maurício Collares, Rafael Montezuma, Roberto Ribeiro, Roberto Teodoro, Vanessa Ribeiro, Vanderson Lima. A trajetória ficou mais suave com a presença de vocês. Às minhas amigas, Renata Thomaz e Clarissa Codá, que me apoiaram e me deram forças nos momentos de pressão e nervosismo. Obrigada por cuidarem tão bem de mim. Às minhas amigas Asun Jiménez, Carolina Mesa, Cristhabel Vasquez, Debora Mondaini, Lis Custódio e Yuri Ki, pelos momentos de descontração e alegria. Às minhas eternas amigas, Welma e Juliana, que mesmo distantes estavam sempre torcendo por mim. Por fim, um agradecimento especial ao Fernando, por tornar meus dias mais alegres, por me confortar em momentos difíceis, e me incentivar a seguir em frente. Acredito que juntos aprendemos a definição de companheirismo. Instituto de Matemática Pura e Aplicada v 2013

6 É através dos estudos, minha filha, que se vence na vida! Antônio e Vivalda

7 Abstract In this work we present some results on minimal and constant mean curvature surfaces in homogeneous 3-manifolds. First, we classify the compact embedded surfaces with constant mean curvature in the quotient of H 2 R by a subgroup of isometries generated by a horizontal translation along horocycles of H 2 and a vertical translation. Moreover, in H 2 R, we construct new examples of periodic minimal surfaces and we prove a multi-valued Rado theorem for small perturbations of the helicoid. In some metric semidirect products, we construct new examples of complete minimal surfaces similar to the doubly and singly periodic Scherk minimal surfaces in R 3. In particular, we obtain these surfaces in the Heisenberg space with its canonical metric, and in Sol 3 with a one-parameter family of non-isometric metrics. After that, we prove a half-space theorem for an ideal Scherk graph Σ M R over a polygonal domain D M, where M is a Hadamard surface with bounded curvature. More precisely, we show that a properly immersed minimal surface contained in D R and disjoint from Σ is a translate of Σ. Finally, based in a joint paper with L. Hauswirth, we prove that if a properly immersed minimal surface in the quotient space H 2 R /G has finite total curvature then its total curvature is a multiple of 2π, and moreover, we understand the geometry of the ends. Here G is a subgroup of isometries generated by a vertical translation and a horizontal isometry in H 2 without fixed points. Keywords: Minimal surfaces, constant mean curvature surfaces, periodic surfaces, uniqueness, finite total curvature. vii

8 Resumo Neste trabalho apresentamos alguns resultados sobre superfícies mínimas e de curvatura média constante em variedades homogêneas tridimensionais. Primeiro, classificamos as superfícies compactas mergulhadas com curvatura média constante no quociente de H 2 R por um subgrupo de isometrias gerado por uma translação horizontal ao longo de horociclos de H 2 e uma translação vertical. Além disso, em H 2 R, construímos novos exemplos de superfícies mínimas periódicas e provamos um teorema de Rado multi-valuado para pequenas perturbações do helicóide. Em alguns produtos semidiretos métricos, contruímos novos exemplos de superfícies mínimas completas similares às superfícies mínimas de Scherk duplamente e simplesmente periódicas em R 3. Em particular, obtemos estas superfícies no espaço de Heisenberg com sua métrica canônica, e em Sol 3 com uma família a um parâmetro de métricas não isométricas. Depois disso, provamos um teorema de semi-espaço para um gráfico de Scherk ideal Σ M R sobre um domínio poligonal D M, onde M é uma superfície de Hadamard com curvatura limitada. Mais precisamente, mostramos que uma superfície mínima propriamente imersa contida em D R e disjunta de Σ é uma translação de Σ. Finalmente, baseado num trabalho em colaboração com L. Hauswirth, provamos que se uma superfície mínima propriamente imersa em H 2 R /G tem curvatura total finita, então sua curvatura total é um múltiplo de 2π e, além disso, entendemos a geometria dos fins. Aqui G é um subgrupo de isometrias gerado por uma translação vertical e uma isometria horizontal de H 2 sem pontos fixos. Palavras-chave: Superfícies mínimas, superfícies com curvatura média conviii

9 stante, superfícies periódicas, unicidade, curvatura total finita. Instituto de Matemática Pura e Aplicada ix 2013

10 Contents Abstract vii Resumo viii Introduction 1 1 Preliminaries Terminology and some basic facts Maximum principle Curvature estimates Flux formula Douglas criterion The Alexandrov problem in a quotient space of H 2 R Introduction Terminology The Alexandrov problem for doubly periodic constant mean curvature surfaces Construction of periodic minimal surfaces Doubly periodic minimal surfaces Vertically periodic minimal surfaces Periodic minimal surfaces invariant by screw motion A multi-valued Rado Theorem Periodic minimal surfaces in semidirect products Introduction Definitions and preliminary results x

11 3.3 A doubly periodic Scherk minimal surface A singly periodic Scherk minimal surface A half-space theorem for ideal Scherk graphs in M R Introduction Definitions and preliminary results Main Result On doubly periodic minimal surfaces in H 2 R with finite total curvature in the quotient space Introduction Definitions and preliminary results Main Results Bibliography 79 Instituto de Matemática Pura e Aplicada xi 2013

12 Introduction One of the most natural and established topics in the differential geometry of surfaces is the global theory of minimal and constant mean curvature surfaces in the space forms R 3, S 3 and H 3. This is a classic field that remains very active nowadays and uses a wide variety of techniques from different subjects, for example, variational calculus, complex analysis, topology, elliptic PDE theory and others. The extension of this classic global theory for the case of immersed surfaces in homogeneous Riemannian three-dimensional manifolds has attracted the attention of many researchers in the last decade. These homogeneous manifolds are the most simple and symmetric Riemannian manifolds that we can consider besides the space forms, together forming the eight 3-dimensional Thurston geometries. This theory is extremely rich, with lots of beautiful examples. Minimal surfaces in H 2 R, for instance, have been used by Collin and Rosenberg [4] to give counterexamples to a well-known conjecture of Schoen and Yau about harmonic diffeomorphisms between the complex plane and the disk. In this work we will present our contributions to the theory of minimal and constant mean curvature surfaces. We will prove some results associated to uniqueness questions, classification problems, construction of new examples of minimal surfaces, halfspace theorems and related themes. Our new results stated here are proved in the papers [18, 35, 36, 37]. In the first chapter, we fix some notations, give some basic definitions, and state well known results that we use in the other chapters. In Chapter 2, we start by proving an Alexandrov type theorem for a quotient space of H 2 R. More precisely, we classify the compact embedded surfaces with constant mean curvature in the quotient of H 2 R by a 1

13 subgroup of isometries generated by a parabolic translation along horocycles of H 2 and a vertical translation. Section 2.4 is devoted to the construction of new examples of periodic minimal surfaces in H 2 R. In Section 2.5, we prove a multi-valued Rado theorem for small perturbations of the helicoid in H 2 R. Chapter 3 focuses on construction of complete minimal surfaces in some metric semidirect products. In Section 3.3, we construct a doubly periodic minimal surface, and in Section 3.4, we construct a singly periodic minimal surface. These surfaces are similar to the doubly and singly periodic Scherk minimal surfaces in R 3. In particular, we obtain these surfaces in the Heisenberg space with its canonical metric, and in Sol 3 with a one-parameter family of non-isometric metrics. In Chapter 4, we prove a half-space theorem for an ideal Scherk graph Σ M R over a polygonal domain D M, where M is a Hadamard surface with bounded curvature. More precisely, we show that a properly immersed minimal surface contained in D R and disjoint from Σ is a translate of Σ. Finally, in Chapter 5, based in a joint work with L. Hauswirth, we prove that if a properly immersed minimal surface in the quotient space H 2 R /G has finite total curvature then its total curvature is a multiple of 2π and, moreover, we understand the geometry of the ends. Here G denotes a subgroup of isometries generated by a vertical translation and a horizontal isometry in H 2 without fixed points. Instituto de Matemática Pura e Aplicada

14 CHAPTER 1 Preliminaries In this chapter we fix notations, give definitions and state some well known results which will be used throughout this work. In Section 1.1, we list some basic definitions as minimal, stable and parabolic surface, and we recall the first and second variational formulae of area. In Section 1.2, we state the maximum principle, which we will use several times in this work. In section 1.3, we state an important result about curvature estimates for stable minimal surfaces. In section 1.4, we give the definition of the Flux formula and state the Flux theorem. Finally, in Section 1.5, we state the Douglas criterion for the existence of a minimal annulus with a certain contour. 1.1 Terminology and some basic facts Let (M, g) be a Riemannian 3-manifold and consider Σ a surface in M. The mean curvature vector of Σ at a point p is defined by H Σ (p) = (A Σ ) p (e i, e i ), i=1 where A Σ denotes the second fundamental form of Σ, and {e 1, e 2 } is an orthonormal basis of T p Σ with respect to the induced metric. Let ν be a local unit normal vector field along Σ around p Σ. The mean curvature of Σ at p with respect to ν is defined by H Σ (p) = H Σ (p), ν(p). 3

15 Remark 1. If there is no ambiguity we will denote the second fundamental form, the mean curvature vector and the mean curvature of Σ only by A, H and H, respectively. Let F : Σ ( ɛ, ɛ) M be a smooth normal variation of compact support of Σ, that is, F (p, 0) = p for all p, F (p, t) = p for all t and p outside some compact set, and the variational vector field X = F (p, 0) is orthogonal t to T p Σ. Denote Σ t = F (p, t). Proposition 1 (First variation formula of area). We have d dt Σ t = 2 H, X dσ, t=0 where Σ t and dσ denote the area of Σ t and the area element of Σ with respect to the induced metric, respectively. d We say that Σ is a minimal surface if dt Σ t t=0 = 0 for every smooth normal variation Σ t of Σ. Hence, Σ is minimal if, anf only if, H 0. Throughout this work we only consider oriented surfaces in oriented Riemannian manifolds. Hence, we can take ν a globally defined unit normal vector field along Σ, and then any variational vector field X of a smooth normal variation Σ t of Σ can be written as X = φν, for some function φ C0 (Σ). Proposition 2 (Second variation formula of area). We have d 2 dt Σ t 2 = Σ φ 2 (Ric(ν, ν) + A 2 )φ 2 dσ, t=0 Σ where Ric denotes the Ricci curvature of M, and Σ φ denotes the gradient of φ on Σ with respect to the induced metric. We say that a minimal surface Σ is stable if d 2 dt Σ t 2 0, t=0 for every smooth normal variation of compact support Σ t of Σ. Notice that if Σ is area-minimizing then Σ is a stable minimal surface, and the condition of stability is equivalent to the first eigenvalue of the Jacobi operator L = Σ + Ric(ν, ν) + A 2 to be nonnegative. Here, Σ denotes the Laplacian on Σ with respect to the induced metric. Let us remark that a simple and useful fact that implies stability is transversality to a Killing field, that is, if a minimal surface is transversal to a Killing field, then it is stable (see, for example, Lemma 2.1 [40]). Σ Instituto de Matemática Pura e Aplicada

16 Definition 1. A surface Σ M is called parabolic if the only functions u : Σ R that satisfy u 0 and u 0 are the constant functions. Otherwise, we say that Σ is hyperbolic. 1.2 Maximum principle A very useful result for studying surfaces with constant mean curvature is the maximum principle. Theorem 1 (Maximum principle). Let Σ 1 and Σ 2 be two constant mean curvature surfaces. Suppose there exists p Σ 1 Σ 2 such that Σ 1 and Σ 2 are tangent at p, and Σ 2 lies in the mean convex side of Σ 1 in a neighborhood of p. Then H 2 H 1, and the equality holds if, and only if, Σ 1 = Σ 2. In particular, the maximum principle implies that if two minimal surfaces are tangent at a point, and one surface lies on one side of the other in a neighborhood of that point, then these two minimal surfaces coincide. For surfaces with boundary we have the following result. Theorem 2 (Boundary maximum principle). Let Σ 1 and Σ 2 be two constant mean curvature surfaces tangent at a point p Σ 1 Σ 2. Suppose that in a neighborhood of p, Σ 1 and Σ 2 can be seen as graphs over the same domain in T p Σ 1 = T p Σ 2, and Σ 2 lies in the mean convex side of Σ 1 in this neighborhood of p. Then H 2 H 1, and the equality holds if, and only if, Σ 1 = Σ Curvature estimates Rosenberg, Souam and Toubiana [50] obtained an estimate for the norm of the second fundamental form of stable H-surfaces in Riemannian 3-manifolds assuming only a bound on the sectional curvature. Their estimate depends on the distance to the boundary of the surface and only on the bound on the sectional curvature of the ambient manifold. More precisely, they proved the following result. Theorem 3 (Rosenberg, Souam and Toubiana, [50]). Let (M, g) be a complete smooth Riemannian 3-manifold of bounded sectional curvature K Λ < +. Then there exists a universal constant C which depends neither on M nor on Λ, satisfying the following: For any immersed stable H-surface Σ in M with trivial normal bundle, and for any p Σ we have A(p) C min{d(p, Σ), π 2 }. Λ Instituto de Matemática Pura e Aplicada

17 On the assumption of uniform curvature estimates we have the following classical result. Proposition 3. Let M be a homogeneous 3-manifold. Let Σ n be an oriented properly embedded minimal surface in N. Suppose there exist c > 0 such that for all n, A Σn c, and a sequence of points {p n } in Σ n such that p n p M. Then there exists a subsequence of Σ n that converges to a complete minimal surface Σ with p Σ. 1.4 Flux formula An important tool for studying minimal and, more generally, constant mean curvature surfaces are the formulae for the flux of appropriately chosen ambient vector fields across the surface. Let u be a function defined in D whose graph is a minimal surface, and consider X = u defined on D, where W 2 = 1 + u 2. For an open domain W U D, and α a boundary arc of U, we define the flux formula across α as F u (α) = X, ν ds; α here α is oriented as the boundary of U and ν is the outer conormal to U along α. Theorem 4 (Flux Theorem). Let U D be an open domain. Then 1. If U is a compact cycle, F u ( U) = If α is a compact arc of U, F u (α) α. 3. If α is a compact arc of U on which u diverges to +, F u (α) = α. 4. If α is a compact arc of U on which u diverges to, F u (α) = α. 1.5 Douglas criterion While a Jordan curve in Euclidean 3-space always bounds a minimal disk, it is generally quite difficult to decide whether a set of several contours is Instituto de Matemática Pura e Aplicada

18 capable of bounding a minimal surface having a prescribed topological type. There is a very important criterion, due to Douglas [9] (see [27], Theorem 2.1, for the case of a general Riemannian manifold), which guarantees the existence of such minimal surface in certain instances. Although the Douglas criterion is quite general, we will only state the particular case that we will use here. For the general statement, see [27]. Theorem 5 (Douglas criterion). Let Γ 1 and Γ 2 be two disjoint Jordan curves. Consider S 1 and S 2 two least area minimal disks with boundary Γ 1 and Γ 2, respectively. If there is an annulus A with boundary Γ 1 Γ 2 such that area(a) area(s 1 ) + area(s 2 ), then there exists a least area minimal annulus with boundary Γ 1 Γ 2. Instituto de Matemática Pura e Aplicada

19 CHAPTER 2 The Alexandrov problem in a quotient space of H 2 R In this chapter we prove an Alexandrov type theorem for a quotient space of H 2 R. More precisely we classify the compact embedded surfaces with constant mean curvature in the quotient of H 2 R by a subgroup of isometries generated by a horizontal translation along horocycles of H 2 and a vertical translation. Moreover, we construct some examples of periodic minimal surfaces in H 2 R and we prove a multi-valued Rado theorem for small perturbations of the helicoid in H 2 R. 2.1 Introduction Alexandrov, in 1962, proved that the only compact embedded constant mean curvature hypersurfaces in R n, H n and S n + are the round spheres. Since then, many people have proved an Alexandrov type theorem in other spaces. For instance, W.T. Hsiang and W.Y. Hsiang [25] showed that a compact embedded constant mean curvature surface in H 2 R or in S 2 + R is a rotational sphere. They used the Alexandrov reflection method with vertical planes in order to prove that for any horizontal direction, there is a vertical plane of symmetry of the surface orthogonal to that direction. To apply the Alexandrov reflection method we need to start with a vertical plane orthogonal to a given direction that does not intersect the surface, and in S 2 R this fact is guaranteed by the hypothesis that the surface is contained in the product of a hemisphere with the real line. We remark that in S 2 R, we know that there are embedded rotational constant mean curvature tori, 8

20 but the Alexandrov problem is not completely solved in S 2 R. In other simply connected homogeneous spaces with a 4-dimensional isometry group (Nil 3, PSL 2 (R), some Berger spheres), we do not know if the solutions to the Alexandrov problem are spheres. In Sol 3, Rosenberg proved that an embedded compact constant mean curvature surface is a sphere [7]. Recently, Mazet, Rodríguez and Rosenberg [29] considered the quotient of H 2 R by a discrete group of isometries of H 2 R generated by a horizontal translation along a geodesic of H 2 and a vertical translation. They classified the compact embedded constant mean curvature surfaces in the quotient space. Moreover, they constructed examples of periodic minimal surfaces in H 2 R, where by periodic we mean a surface which is invariant by a non-trivial discrete group of isometries of H 2 R. We also consider periodic surfaces in H 2 R. The discrete groups of isometries of H 2 R we consider are generated by a horizontal translation ψ along horocycles c(s) of H 2 and/or a vertical translation T (h) for some h > 0. In the case the group is the Z 2 subgroup generated by ψ and T (h), the quotient space M = H 2 R /[ψ, T (h)] is diffeomorphic to T 2 R, where T 2 is the 2-torus. Moreover, M is foliated by the family of tori T(s) = c(s) R /[ψ, T (h)] which are intrinsically flat and have constant mean curvature 1/2. In this quotient space M, we prove an Alexandrov type theorem. Moreover, we consider a multi-valued Rado theorem for small perturbations of the helicoid. Rado s theorem (see [47]) is one of the fundamental results of minimal surface theory. It is connected to the famous Plateau problem, and states that if Ω R 2 is a convex subset and Γ R 3 is a simple closed curve which is graphical over Ω, then any compact minimal surface Σ R 3 with Σ = Γ must be a disk which is graphical over Ω, and then unique, by the maximum principle. In [8], Dean and Tinaglia proved a generalization of Rado s theorem. They showed that for a minimal surface of any genus whose boundary is almost graphical in some sense, the minimal surface must be graphical once we move sufficiently far from the boundary. In our work, we consider this problem for minimal surfaces in H 2 R whose boundary is a small perturbation of the boundary of a helicoid, and we prove that the solution to the Plateau problem is the only compact minimal disk with that boundary (see Theorem 7). This chapter is organized as follows. In section 2.2, we introduce some notation used in this chapter. In Section 2.3, we classify the compact embedded constant mean curvature surfaces in the space M, that is, we prove an Alexandrov type theorem for doubly periodic H-surfaces (see Theorem 6). In section 2.4, we construct some examples of periodic minimal surfaces in H 2 R. In section 2.5, we prove a multi-valued Rado theorem for small Instituto de Matemática Pura e Aplicada

21 perturbations of the helicoid (see Theorem 7). 2.2 Terminology Throughout this chapter, the Poincaré disk model is used for the hyperbolic plane, that is, H 2 = {(x, y) R 2 x 2 + y 2 < 1} 4 with the hyperbolic metric g 1 = g (1 x 2 y 2 ) 2 0, where g 0 is the Euclidean metric in R 2. In this model, the asymptotic boundary H 2 of H 2 is identified with the unit circle. Consequently, any point in the closed unit disk is viewed as either a point in H 2 or a point in H 2. We denote by 0 the origin of H 2. In H 2 we consider γ 0, γ 1 the geodesic lines {x = 0}, {y = 0}, respectively. For j = 0, 1, we denote by Y j the Killing vector field whose flow (φ l ) l ( 1,1) is given by hyperbolic translation along γ j with φ l (0) = (l sin πj, l cos πj) and (sin πj, cos πj) as attractive point at infinity. We call (φ l ) l ( 1,1) the flow of Y j even though the family (φ l ) l ( 1,1) is not parameterized at the right speed. We denote by π : H 2 R H 2 the vertical projection and we write t for the height coordinate in H 2 R. In what follows, we will often identify the hyperbolic plane H 2 with the horizontal slice {t = 0} of H 2 R. The vector fields Y j, j = 0, 1, and their flows naturally extend to horizontal vector fields and their flows in H 2 R. Consider any geodesic γ that limits to the point p 0 H 2 at infinity parametrized by arc length. Let c(s) denote the horocycle in H 2 tangent to H 2 at p 0 that intersects γ at γ(s). Given two points p, q c(s), we denote by ψ : H 2 R H 2 R the parabolic translation along c(s) such that ψ(p) = q. We write pq to denote the geodesic arc between the two points p, q of H 2 R. 2.3 The Alexandrov problem for doubly periodic constant mean curvature surfaces Take two points p, q in a horocycle c(s), and let ψ be the parabolic translation along c(s) such that ψ(p) = q. We have ψ(c(s)) = c(s) for all s. Consider G the Z 2 subgroup of isometries of H 2 R generated by ψ and a vertical translation T (h), for some positive h. We denote by M the quotient of H 2 R by G. The manifold M is diffeomorphic but not isometric to T 2 R and is foliated by the family of tori T(s) = (c(s) R)/G, s R, which are Instituto de Matemática Pura e Aplicada

22 intrinsically flat and have constant mean cuvature 1/2. Thus the tori T(s) are examples of compact embedded constant mean curvature surfaces in M. We have the following answer to the Alexandrov problem in M. Theorem 6. Let Σ M be a compact immersed surface with constant mean curvature H. Then H 1 2. Moreover, 1. If H = 1, then Σ is a torus T(s), for some s; 2 2. If H > 1 and Σ is embedded, then Σ is either the quotient of a rotational 2 sphere, or the quotient of a vertical unduloid (in particular, a vertical cylinder over a circle). Proof. Let Σ be a compact immersed surface in M with constant mean curvature H. As Σ is compact, there exist s 0 s 1 R such that Σ is between T(s 0 ) and T(s 1 ), and it is tangent to T(s 0 ), T(s 1 ) at points q, p, respectively, as illustrated in Figure 2.1. Figure 2.1: Σ M. For s < s 0, the torus T(s) does not intersect Σ, and Σ stays in the mean convex region bounded by T(s). By comparison at q, we conclude that H 1 2. If H = 1 2, then by the maximum principle, Σ is the torus T(s 0), and we have proved the first part of the theorem. To prove the last part, suppose Σ is embedded and consider the quotient space M = H 2 R /[T (h)], which is diffeomorphic to H 2 S 1. Take a connected component Σ of the lift of Σ to M, and denote by c(s) the surface c(s) S 1. Observe that c(s) is the lift of T(s) to M. Moreover, let us consider two points p, q Σ whose projections in M are the points p, q, respectively. It is easy to prove that Σ separates M. In fact, suppose by contradiction this is not true, then we can consider a geodesic arc α : ( ɛ, ɛ) M such that α(0) Σ, α (0) T Σ and we can join the points α( ɛ), α(ɛ) by a curve that does not intersect Σ, hence we obtain a Jordan curve, which we Instituto de Matemática Pura e Aplicada

23 still call α, whose intersection number with Σ is 1 modulo 2. Notice that the distance between Σ and c(s 0 ) is bounded. Since we can homotop α so it is arbitrarily far from c(s 0 ), we conclude that a translate of α does not intersect Σ, contradicting the fact that the intersection number of α and Σ is 1 modulo 2. Thus Σ does separate M. Let us call A the mean convex component of M \ Σ with boundary Σ and B the other component. Hence M \ Σ = A B. Let γ be a geodesic in H 2 that limits to p 0 H 2, γ(+ ) = p 0 (the point where the horocycles c(s) are centered) and let us assume that γ intersects Σ in at least two points. Consider (l t ) t R the family of geodesics in H 2 orthogonal to γ and denote by P (t) the totally geodesic vertical annulus l t S 1 of M = H 2 S 1 (see Figure 2.2). Since Σ is a lift of the compact surface Σ, it stays in the region between c(s 0 ) and c(s 1 ), and the distance from any point of Σ to c(s 0 ) and to c(s 1 ) is uniformly bounded. Figure 2.2: The family of totally geodesic annuli P (t). By our choice of γ, the ends of each P (t) are outside the region bounded by c(s), hence P (t) Σ is compact for all t. Moreover, for t close to, P (t) is contained in B and P (t) Σ is empty. Then start with t close to and let t increase until a first contact point between Σ and some vertical annulus, say P (t 0 ). In particular, we know the mean curvature vector of Σ does not point into t t 0 P (t). Continuing to increase t and starting the Alexandrov reflection procedure for Σ and the family of vertical totally geodesic annuli P (t), we get a first contact point between the reflected part of Σ and Σ, for some t 1 R. Observe that this first contact point occurs because we are assuming that the geodesic γ intersects Σ in at least two points. Then Σ is symmetric with respect to P (t 1 ). As Σ ( t 0 t t 1 P (t) ) is Instituto de Matemática Pura e Aplicada

24 compact, then Σ is compact. Hence, given any horizontal geodesic α we can apply the Alexandrov procedure with the family of totally geodesic vertical annuli Q(t) = l t S 1, where ( l t ) t R is the family of horizontal geodesics orthogonal to α, and we obtain a symmetry plane for Σ. Hence we have shown that if some geodesic that limits to p 0 intersects Σ in two or more points, then Σ lifts to a rotational cylindrically bounded surface Σ in H 2 R. If Σ is not compact then Σ is a vertical unduloid, and if Σ is compact we know, by Hsiang-Hsiang s theorem [25], Σ is a rotational sphere. Therefore, we have proved that in this case Σ M is either the quotient of a rotational sphere or the quotient of a vertical unduloid. Now to finish the proof let us assume that every geodesic that limits to p 0 intersects Σ in at most one point. In particular, the geodesic β that limits to p 0 and passes through p c(s 1 ) intersects Σ only at p. Write β to denote the arc of β between β( ) and p (see Figure 2.3). Figure 2.3: Geodesic β. As β Σ = { p}, we have β Σ = and then β B, since Σ separates M. Hence at the point p Σ c(s 1 ), the mean curvature vectors of Σ and c(s 1 ) point to the mean convex side of c(s 1 ) and Σ lies on the mean concave side of c(s 1 ), then by comparison we get H 1. But we already know that 2 H 1. Hence H = 1 and Σ = c(s ), by the maximum principle. Therefore, in this case we conclude Σ = T(s 1 ). Remark 2. Note that a vertical unduloid, contained in a cylinder D R and invariant by a vertical translation T (l) in H 2 R, passes to the quotient space M = H 2 R /[ψ, T (h)] as an embedded surface if the quotient of D is embedded and the number l is a multiple of h. Analogously, a rotational sphere of height l contained in a cylinder D R in H 2 R passes to the Instituto de Matemática Pura e Aplicada

25 quotient as an embedded surface if l < h and the quotient of D is embedded in M. 2.4 Construction of periodic minimal surfaces In this section we are interested in constructing some new examples of periodic minimal surfaces in H 2 R invariant by a subgroup of isometries, which is either isomorphic to Z 2, or generated by a vertical translation, or generated by a screw motion. In fact, we only consider subgroups generated by a parabolic translation ψ along a horocycle and/or a vertical translation T (h), for some h > 0. Periodic minimal surfaces in R 3 have received great attention since Riemann, Schwarz, Scherk (and many others) studied them. They also appear in the natural sciences. In [33], Meeks and Rosenberg proved that a periodic properly embedded minimal surface of finite topology (in R 3 /G, G a discrete group of isometries acting properly discontinuously on R 3, G (1)) has finite total curvature and the ends are asymptotic to standard ends (planar, catenoidal, or helicoidal). In a joint paper with Hauswirth [18], we consider the same study for periodic minimal surfaces in H 2 R. The first step is to understand what are the possible models for the ends in the quotient. This is one reason to construct examples Doubly periodic minimal surfaces In H 2 consider two geodesics α, β that limit to the same point at infinity, say α( ) = p 0 = β( ). Denote B = α(+ ) and D = β(+ ). Take a geodesic γ contained in the region bounded by α and β that limits to the same point p 0 at infinity. Parametrize these geodesics so that α(t) B, β(t) D and γ(t) p 0 when t +. Fix h > π and consider the following Jordan curve: Γ t = (α(t), 0), (γ(t), 0) (α(t), 0), (α(t), h) (β(t), 0), (γ(t), 0) (β(t), 0), (β(t), h) (α(t), h), (γ(t), h) (β(t), h), (γ(t), h) as illustrated in Figure 4.1. Consider a least area embedded minimal disk Σ t with boundary Γ t. Let Y be the Killing field whose flow (φ l ) l ( 1,1) is given by translation along the geodesic γ. Notice that Γ t is transversal to the Killing field Y. Hence given any geodesic γ orthogonal to γ, we can use the Alexandrov reflection technique with the foliation of H 2 R by the vertical planes (φ l ( γ)) l ( 1,1) to show that Instituto de Matemática Pura e Aplicada

26 Figure 2.4: Curve Γ t. Σ t is a Y -Killing graph. In particular, Σ t is stable and unique (see Lemma 2.1 in [40]). This gives uniform curvature estimates for Σ t0 for points far from the boundary (see Main Theorem in [50]). Rotating Σ t by angle π around the geodesic arc (α(t), 0), (γ(t), 0) gives a minimal surface that extends Σ t, has int(α(t), 0), (γ(t), 0) in its interior, and is still a Y -Killing graph. Thus we get uniform curvature estimates for Σ t in a neighborhood of (α(t), 0), (γ(t), 0). This is also true for the three other horizontal geodesic arcs in Γ t. Observe that for any t, Σ t stays in the half-space determined by BD R that contains Γ t, by the maximum principle. As h > π, we can use as a barrier the minimal surface S h H 2 (0, h) which is a vertical bigraph with respect to the horizontal slice {t = h}. 2 The surface S h is invariant by translations along the horizontal geodesic γ 0 = {x = 0} and its asymptotic boundary is (τ {0}) (0, 1, 0)(0, 1, h) (τ {h}) (0, 1, 0)(0, 1, h), where τ = H 2 {x > 0}. For more details about the surface S h, see [29, 30, 51]. For l close to 1, the translated surface φ l (S h ) does not intersect Σ t. Hence the surface Σ t is contained between φ l (S h ) and BD R. Notice that when t +, Γ t converges to Γ, where Γ = (α {0}) (β {0}) (α {h}) (β {h}) (D, 0)(D, h) (B, 0)(B, h). Therefore, as we have uniform curvature estimates and barriers at infinity, there exists a subsequence of Σ t that converges to a minimal surface Σ, where Σ lies in the region of H 2 [0, h] bounded by α R, β R, BD R and φ l (S h ); with boundary Σ = Γ. Hence the surface obtained by reflection in all horizontal boundary geodesics of Σ is invariant by ψ 2 and T (2h), where ψ is the horizontal translation along horocycles that sends α to β. Moreover, this surface in the quotient space Instituto de Matemática Pura e Aplicada

27 H 2 R /[ψ 2, T (2h)] is topologically a sphere minus four points. Two ends are asymptotic to vertical planes and two are asymptotic to horizontal planes (cusps), all of them with finite total curvature. Proposition 4. There exists a doubly periodic minimal surface (invariant by horizontal translations along a horocycle and by a vertical translation) such that, in the quotient space, this surface is topologically a sphere minus four points, with two ends asymptotic to vertical planes and two asymptotic to horizontal planes, all of them with finite total curvature Vertically periodic minimal surfaces Take α any geodesic in H 2 {0}. For h > π, consider the vertical segment α( ) [0, 2h], and a point p H 2, p α( ), α(+ ). For some small ɛ > 0, consider the asymptotic vertical segment joining (p, ɛ) and (p, h + ɛ). Now, connect (p, ɛ) to (α( ), 0) and (p, h + ɛ) to (α( ), 2h) by curves in H 2 R, whose tangent vectors are never horizontal or vertical, and so that the resulting curve Γ is differentiable. Also, consider the horizontal geodesic β connecting p to α(+ ). Parametrize α by arc length, and consider γ a geodesic orthogonal to α passing through α(0). Let us denote by d(t) the equidistant curve to γ in a distance t that intersects α at α(t). For each t consider a curve Γ t contained in the plane d(t) R with endpoints (α(t), 0) and (α(t), 2h) such that Γ t is contained in the region R bounded by α R, β R, H 2 {0} and H 2 {2h} with the properties that its tangent vectors do not point in the horizontal direction and Γ t converges to Γ when t. In particular, Γ t is transversal to the Killing field Y whose flow (φ l ) l ( 1,1) is given by translation along the geodesic γ. Write α t to denote the vertical segment α(t) [0, 2h] (see Figure 2.5). Figure 2.5: Curves Γ n and Γ. Instituto de Matemática Pura e Aplicada

28 For each n, let Σ n be the solution to the Plateau problem with boundary Γ n (α([ n, n]) {0}) (α([ n, n]) {2h}) α n. By our choice of the curves Γ t, the boundary Σ n is transverse to the Killing field Y. Using the foliation of H 2 R by the vertical planes φ l (α), l ( 1, 1), the Alexandrov reflection technique shows that Σ n is a Y -Killing graph. In particular, it is unique and stable [40], and we have uniform curvature estimates far from the boundary [50]. When we apply the rotation by angle π around α {0} to the minimal surface Σ n, we get another minimal surface which extends Σ n, is still a Y-Killing graph and has int(α([ n, n]) {0}) in its interior. Hence we obtain uniform curvature estimates for Σ n in a neighborhood of α([ n, n]) {0}. This is also true for α([ n, n]) {2h} and α n. Observe that Σ n is contained in the region R, for all n. By our choice of Γ, for each q Γ, we can consider two translations of the minimal surfaces S h (considered in the last section) that pass through q so that one of them has asymptotic boundary under Γ, the other one has asymptotic boundary above Γ and their intersection with Γ is just the point q considered or is the whole vertical segment (p, ɛ)(p, h + ɛ). Hence, the envelope of the union of all these translated surfaces S h forms a barrier to Σ n, for all n. Then, as we have uniform curvature estimates and barriers at infinity, we conclude that there exists a subsequence of Σ n that converges to a minimal surface Σ with (α(+ ) [0, 2h]) Γ = Σ, and then Σ = Γ (α {0}) (α {2h}) (α(+ ) [0, 2h]). Therefore, the surface obtained by reflection in all horizontal boundary geodesics of Σ is a vertically periodic minimal surface invariant by T (4h). In the quotient space this minimal surface has two ends; one is asymptotic to a vertical plane and has finite total curvature, while the other one is topologically an annular end and has infinite total curvature. Proposition 5. There exists a singly periodic minimal surface (invariant by a vertical translation) such that, in the quotient space, this surface has two ends, one end is asymptotic to a vertical plane and has finite total curvature, while the other one is topologically an annular end and has infinite total curvature Periodic minimal surfaces invariant by screw motion Now we construct some examples of periodic minimal surfaces invariant by a screw motion, that is, invariant by a subgroup of isometries generated by the composition of a horizontal translation with a vertical translation. Instituto de Matemática Pura e Aplicada

29 Consider two geodesics α, β in H 2 that limit to the same point at infinity, say α(+ ) = p 0 = β(+ ). For h > π, consider a smooth curve Γ contained in the asymptotic boundary of H 2 R, connecting (α( ), 2h) to (β( ), 0) and such that its tangent vectors are never horizontal or vertical. Also, take a point p H 2 in the halfspace determined by β R that does not contain α. For some small ɛ > 0, consider the asymptotic vertical segment joining (p, ɛ) and (p, h + ɛ). Now, connect (p, ɛ) to (p 0, 0) and (p, h + ɛ) to (p 0, 2h) by curves in H 2 R whose tangent vectors are never horizontal or vertical, and such that the resulting curve Γ is differentiable. Parametrize α by arc length, and consider γ a geodesic orthogonal to α passing through α(0). Let us denote by d(t) the equidistant curve to γ in a distance t that intersects α at α(t). For each t, s consider two curves Γ t and Γ s contained in the plane d(t) R and d(s) R, respectively, with the properties that their tangent vectors are never horizontal, Γ t joins (α(t), 2h) to (β(t), 0), Γ s joins (α(s), 2h) to (β(s), 0), Γ t converges to Γ when t +, Γ s converges to Γ when s, and both curves are contained in the region R bounded by α R, θ R, H 2 {0} and H 2 {2h}, where θ is the geodesic with endpoints p and β( ) (see Figure 2.6). Figure 2.6: Curves Γ t, Γ s, Γ and Γ. For each n, let Σ n be the solution to the Plateau problem with boundary Γ n (α([ n, n]) {2h}) Γ n (β([ n, n]) {0}). The surface Σ n is contained in the region R. As in the previous section, we can show that Σ n is a Killing graph, then it is stable, unique and we have uniform curvature estimates far from the boundary. Rotating Σ n by angle π around the geodesic α {2h} we get a minimal surface which extends Σ n, is still a Killing graph, and has int(α([ n, n]) {2h}) in its interior. Hence we get uniform curvature estimates for Σ n in a neighborhood of α([ n, n]) {2h}. This is also true for β([ n, n]) {0}. Thus when n +, there exists a subsequence of Σ n that converges to a minimal surface Σ with Γ Γ Σ n. Using the same Instituto de Matemática Pura e Aplicada

30 argument as before with suitable translations of the surface S h as barriers, we conclude that in fact Σ = Γ Γ, and then Σ = Γ (α {2h}) (β {0}) Γ. The surface obtained by reflection in all horizontal boundary geodesics of Σ is a minimal surface invariant by ψ 2 T (4h), where ψ is the horizontal translation along horocycles that sends α to β. There are two annular embedded ends in the quotient, each of infinite total curvature. Proposition 6. There exists a minimal surface invariant by a screw motion such that, in the quotient space, this minimal surface has two annular embedded ends, each one of infinite total curvature. Now we will construct another interesting example of a periodic minimal surface invariant by a screw motion. Denote by γ 0, γ 1 the geodesic lines {x = 0}, {y = 0} in H 2, respectively. Let c be a horocycle orthogonal to γ 1, and consider p, q c equidistant points to γ 1. Take α, β geodesics which limit to p 0 = (1, 0) = γ 1 (+ ) and pass through p, q, respectively. Fix ɛ > 0 and h > π. Consider the points A = α( t 0 ), C = α(t 0 ), B = β( t 0 ), D = β(t 0 ), and let us consider the following Jordan curve (see Figure 2.7): Γ t0 = (α([ t 0, t 0 ]) { ɛ}) (C, ɛ)(d, 0) (β([ t 0, t 0 ]) {0}) (α([ t 0, t 0 ]) {h}) (C, h)(d, h + ɛ) (β([ t 0, t 0 ]) {h + ɛ}) (A, ɛ)(a, h) (B, 0)(B, h + ɛ). Figure 2.7: Curve Γ t0. We consider a least area embedded minimal disk Σ t0 with boundary Γ t0. Denote by Y 1 the Killing vector field whose flow (φ l ) l ( 1,1) gives the hyperbolic translation along γ 1 with φ l (0) = (l, 0) and p 0 as attractive point Instituto de Matemática Pura e Aplicada

31 at infinity. As Γ t0 is transversal to the Killing field Y 1, we can prove, using the maximum principle, that Σ t0 is a Y 1 -Killing graph with convex boundary, in particular, Σ t0 is stable and unique [40]. This yields uniform curvature estimates far from the boundary [50]. Rotating Σ t0 by angle π around the geodesic arc α([ t 0, t 0 ]) { ɛ} gives a minimal surface that extends Σ t0, has int(α([ t 0, t 0 ]) { ɛ}) in its interior, and is still a Y 1 -Killing graph. Thus we get uniform curvature estimates for Σ t0 in a neighborhood of α([ t 0, t 0 ]) { ɛ}. This is also true for the three other horizontal geodesic arcs in Γ t0. Write F = α( ), G = β( ). Observe that, by the maximum principle, for any t 0, Σ t0 stays in the halfspace determined by F G R that contains Γ t0. Since h > π, we can consider the minimal surface S h (considered in Section 2.4.1) as a barrier. For l close to 1, the translated surface φ l (S h ) does not meet Σ t0. The surface Σ t0 is contained between φ l (S h ) and F G R. When t 0 +, Γ t0 converges to Γ, where Γ = (α { ɛ}) (p 0, ɛ)(p 0, 0) (β {0}) (α {h}) (p 0, h)(p 0, h + ɛ) (β {h + ɛ}) (F, ɛ)(f, h) (G, 0)(G, h + ɛ). Using the maximum principle, we can prove that Σ t is contained between φ l (S h ) and F G R, for all t > t 0. Therefore, there exists a subsequence of the surfaces Σ t that converges to a minimal surface Σ, where Σ lies in the region between H 2 { ɛ} and H 2 {h+ɛ} bounded by α R, β R, F G R and φ l (S h ); and has boundary Σ = Γ. Hence the surface obtained by reflection in all horizontal boundary geodesics of Σ is invariant by ψ 2 T (2(h + ɛ)), where ψ is the horizontal translation along horocycles that sends α to β. Moreover, this surface in the quotient space has two vertical ends and two helicoidal ends, each one of finite total curvature. Proposition 7. There exists a minimal surface invariant by a screw motion such that, in the quotient space, this minimal surface has four ends. Two vertical ends and two helicoidal ends, all of them with finite total curvature. 2.5 A multi-valued Rado Theorem The aim of this section is to prove a multi-valued Rado theorem for small perturbations of the helicoid. Recall that Rado s theorem says that minimal surfaces over a convex domain with graphical boundaries must be disks Instituto de Matemática Pura e Aplicada

32 which are themselves graphical. We will prove that for certain small perturbations of the boundary of a (compact) helicoid there exists only one compact minimal disk with that boundary. By a compact helicoid we mean the intersection of a helicoid with certain compact regions in H 2 R. The idea here originated in the work of Hardt and Rosenberg [16]. We will apply this multi-valued Rado theorem to construct an embedded minimal surface in H 2 R whose boundary is a small perturbation of the boundary of a complete helicoid. Consider Y the Killing field whose flow φ θ, θ [0, 2π), is given by rotations around the z-axis. For some 0 < c < 1, let D = {(x, y) H 2 ; x 2 + y 2 c}. Take a helix h 0 of constant pitch contained in a solid cylinder D [0, d], so that the vertical projection of h 0 over H 2 {0} is D, and the endpoints of h 0 are in the same vertical line. Let us denote by Γ 0 the Jordan curve which is the union of h 0, the two horizontal geodesic arcs joining the endpoints of h 0 to the z-axis, and the part of the z-axis. Call H the compact part of the helicoid that has Γ 0 as its boundary. We know that H is a minimal surface transversal to the Killing field Y at the interior points. Take θ < π/4, and consider H 1 = φ θ (H) and H 2 = φ θ (H). Hence H 1, H 2 are two compact helicoids with boundary H 1 = φ θ (Γ 0 ), H 2 = φ θ (Γ 0 ). Consider h a small smooth perturbation of the helix h 0 with fixed endpoints such that h is transversal to Y and h is contained in the region between φ θ (h 0 ) and φ θ (h 0 ) in D [0, d]. Call Γ the Jordan curve which is the union of h, the two horizontal geodesic arcs and a part of the z-axis, hence Γ = (Γ 0 \ h 0 ) h (see Figure 2.8). Figure 2.8: Curve Γ. Denote by R the convex region bounded by H 1 and H 2 in the solid cylinder D [0, d]. The Jordan curve Γ is contained in the simply connected region R which has mean convex boundary. Then we can consider the solution to the Plateau problem in this region R, and we get a compact minimal disk H contained in R with boundary H = Γ. Proposition 8. Under the assumptions above, H is transversal to the Killing Instituto de Matemática Pura e Aplicada

33 field Y at the interior points. Moreover, the family (φ θ (H)) θ [0,2π) foliates D [0, d] \ {z-axis}. Proof. As H is a disk, we already know that each integral curve of Y intersects H in at least one point. Observe that φ π/2 (R) R \ {z-axis} = and, in particular, φ π/2 (H) H \ {z-axis} =. Moreover, notice that the tangent plane of φ π/2 (H) never coincides with the tangent plane of H along the z-axis; at each point of the z-axis the surfaces are in disjoint sectors. So as one decreases t from π/2 to 0, the surfaces φ t (H) and H have only the z-axis in common and they are never tangent along the z-axis. More precisely, as t decreases, t > 0, there can not be a first interior point of contact between the two surfaces by the maximum principle. Also there can not be a point on the z-axis which is a first point of tangency of the two surfaces for t > 0, by the boundary maximum principle. Thus the surfaces φ t (H) and H have only the z-axis in common for t > 0. The same argument works for π/2 t < 0. Therefore the surfaces φ t (H) foliate D [0, d] \ {z-axis}, for t [0, 2π). In particular, we have concluded that each integral curve of Y intersects H in exactly one point. Denote by R 2 the region in R bounded by H and H 2, and denote by N the unit normal vector field of H pointing toward R 2. As each integral curve of Y intersects H in exactly one point, we have N, Y 0 on H. As N, Y is a Jacobi function on the minimal surface H, we conclude that necessarily N, Y > 0 in inth. Therefore, H is transversal to the Killing field Y at the interior points. Theorem 7 (A multi-valued Rado Theorem). Under the assumptions above, H is the unique compact minimal disk with boundary Γ. Proof. Set Γ θ = φ θ (Γ) and H θ = φ θ (H), so H θ is a minimal disk with H θ = Γ θ. By Proposition 8, the family (H θ ) θ [0,2π) gives a foliation of the region D [0, d] \ {z-axis}. Let M H be another compact minimal disk with boundary Γ. We will analyse the intersection between M and each H θ. First, observe that M H θ for all θ and by the maximum principle M D [0, d]. Fix θ 0. Given q H θ0 M, then either q intm or q Γ = M. Suppose q intm. If the intersection is transversal at q, then in a neighborhood of q we have that H θ0 M is a simple curve passing through q. If we let θ 0 vary a little, we see in M a foliation as in Figure 2.9 (a). On the other hand, if M is tangent to H θ0 at q, as the intersection of any two minimal surfaces is locally given by an n-prong singularity, that is, Instituto de Matemática Pura e Aplicada